A branching process model where offspring distributions depend on the threshold as well as on the population size is introduced. Behaviour of such models is related to the behaviour of the corresponding deterministic models, whose behaviour is known from the chaos theory. Asymptotic behaviour of such branching processes is obtained when the population threshold is large. If the initial population size is comparable to the threshold then the size of the nth generation relative to the threshold has a normal distribution with the mean being the nth iterate of the one-step mean function. If the initial population size is negligible when compared to the threshold and the offspring distributions converge then the size of any fixed generation approaches that of a size-dependent branching process. These results are supported by a simulation study.